Chapter 2: Polynomials

Important questions — the ones the board asked most (20)

1. Q1Short answer3 marksAsked: 2023, 2025

   Find the zeroes of the polynomial p(x) = 4x^2 - 4x - 3 and verify the relationship between its coefficients and zeroes.

   

   Medha Ma’am

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   Model answer · zeroes of quadratic polynomial

   1. 1.Given p(x) = 4x^2 – 4x – 3. We factorise it by splitting the middle term: p(x) = 4x^2 – 6x + 2x – 3 = 2x(2x – 3) + 1(2x – 3) = (2x + 1)(2x – 3) So the value of p(x) is zero when 2x + 1 = 0 or 2x – 3 = 0, that is, when x = –1/2 or x = 3/2. Therefore, the zeroes of p(x) are –1/2 and 3/2.1 mark
   2. 2.Comparing p(x) with ax^2 + bx + c, we get a = 4, b = –4 and c = –3. Sum of zeroes = –1/2 + 3/2 = 2/2 = 1 –(Coefficient of x)/(Coefficient of x^2) = –(–4)/4 = 1 So, sum of zeroes = –(Coefficient of x)/(Coefficient of x^2). Verified.1 mark
   3. 3.Product of zeroes = (–1/2) × (3/2) = –3/4 Constant term/(Coefficient of x^2) = –3/4 So, product of zeroes = Constant term/(Coefficient of x^2). Verified. Hence the relationship between the zeroes and the coefficients is verified.1 mark

   • Show the splitting of the middle term, 4x^2 – 6x + 2x – 3, before writing (2x + 1)(2x – 3).
   • Write the formula "–(Coefficient of x)/(Coefficient of x^2)" with –(–4)/4 = 1 next to the sum of zeroes.
   • Write "Constant term/(Coefficient of x^2)" = –3/4 next to the product of zeroes, and write "Verified" after each one.

   Marks split as in the board’s published marking scheme

2. Q2Short answer3 marksAsked: 2023, 2025

   Find the zeroes of the polynomial p(x) = 6x^2 + 13x - 5 and verify the relationship between its zeroes and the coefficients.

   

   Medha Ma’am

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   Model answer · Zeroes and coefficients, quadratic

   1. 1.Given the quadratic polynomial p(x) = 6x^2 + 13x - 5, we factorise it by the method of splitting the middle term: p(x) = 6x^2 + 15x - 2x - 5 = 3x(2x + 5) - 1(2x + 5) = (3x - 1)(2x + 5)1 mark
   2. 2.So the value of p(x) is zero when 3x - 1 = 0 or 2x + 5 = 0, which means x = 1/3 or x = -5/2. Therefore, the zeroes of p(x) are 1/3 and -5/2.1 mark
   3. 3.Verification: comparing p(x) with ax^2 + bx + c, we get a = 6, b = 13 and c = -5. Sum of zeroes = 1/3 + (-5/2) = (2 - 15)/6 = -13/6 -(Coefficient of x)/(Coefficient of x^2) = -b/a = -13/6 Product of zeroes = (1/3) × (-5/2) = -5/6 Constant term/(Coefficient of x^2) = c/a = -5/6 Both sides match, so the relationship between the zeroes and the coefficients is verified.1 mark

   • Write out the splitting step "6x^2 + 15x - 2x - 5" before you write the factors (3x - 1)(2x + 5).
   • For the sum of zeroes, write both -13/6 and "-(Coefficient of x)/(Coefficient of x^2)" so the examiner can see they are equal.

   Marks split as in the board’s published marking scheme

3. Q3Very short answer2 marksAsked: 2023, 2026

   Find the zeroes of the quadratic polynomial x^2 + 7x + 10, and verify the relationship between the zeroes and its coefficients.

   

   Medha Ma’am

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   Hear it from Medha Ma’am · 90 s

   Model answer · zeroes and coefficients, quadratic

   1. 1.Given quadratic polynomial: x^2 + 7x + 10. By splitting the middle term: x^2 + 7x + 10 = x^2 + 2x + 5x + 10 = x(x + 2) + 5(x + 2) = (x + 2)(x + 5). The value of x^2 + 7x + 10 is zero when x + 2 = 0 or x + 5 = 0, i.e., when x = –2 or x = –5. So, the zeroes of the polynomial are –2 and –5.1 mark
   2. 2.Here, coefficient of x^2 = 1, coefficient of x = 7, constant term = 10. Sum of zeroes = (–2) + (–5) = –7 = –7/1 = –(Coefficient of x)/(Coefficient of x^2). Hence verified.0.5 marks
   3. 3.Product of zeroes = (–2) × (–5) = 10 = 10/1 = (Constant term)/(Coefficient of x^2). Hence, the relationship between the zeroes and the coefficients is verified.0.5 marks

   • Write the factor form (x + 2)(x + 5) first, then state that the zeroes are –2 and –5.
   • For the sum, write the full chain "–7 = –7/1 = –(Coefficient of x)/(Coefficient of x^2)", including the minus sign in front of the coefficient of x.
   • For the product, write "10 = 10/1 = (Constant term)/(Coefficient of x^2)" so the relationship is clearly checked.

   Marks split as in the board’s published marking scheme

4. Q4Case-based4 marksAsked: 2026

   36. During a theatre drama, a backdrop of building arches was used. The shape of the curve shown below can be represented by the polynomial p(x) = - x^2 + 2x + 8, where x is the length (in feet) on stage level. Based on the figure given above, answer the following questions : (i) Determine the height of the arch. (ii) (a) Find zeroes of the polynomial p(x). Which points on the graph represent the zeroes? OR (ii) (b) Find the span of the arch on the stage floor. (iii) Write the coordinates of the point of intersection of the above curve with the y-axis.

   How the model answer begins

   1. (i) The arch is the curve y = p(x), where p(x) = – x² + 2x + 8. Its highest point is midway between the points where it meets the stage floor, x = – 2 and x = 4, so it is at x = 1. Height of the arch = p(1) = – (1)² + 2…

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5. Q5Case-based4 marksAsked: 2026

   Case Study - 3 38. During a theatre drama, a backdrop of building arches was used. The shape of the curve shown below can be represented by the polynomial p(x) = - x^2 + 2x + 8, where x is the length (in feet) on stage level. Based on the figure given above, answer the following questions : (i) Determine the height of the arch. (ii) (a) Find zeroes of the polynomial p(x). Which points on the graph represent the zeroes? OR (ii) (b) Find the span of the arch on the stage floor. (iii) Write the coordinates of the point of intersection of the above curve with the y-axis.

   How the model answer begins

   1. The height of the arch is found by calculating the maximum value (vertex) of the polynomial p(x) = -x^2 + 2x + 8, since the arch is open downwards and its highest point is the vertex of the parabola. p(x) = -x^2 + 2x +…

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6. Q6Case-based4 marksAsked: 2024

   A ball is thrown in the air so that t seconds after it is thrown, its height h metre above its starting point is given by the polynomial h = 25t - 5t^2. Observe the graph of the polynomial and answer the following questions : (i) Write zeroes of the given polynomial. (ii) Find the maximum height achieved by ball. (iii) (a) After throwing upward, how much time did the ball take to reach to the height of 30 m ? OR (iii) (b) Find the two different values of t when the height of the ball was 20 m.

   How the model answer begins

   1. (i) The zeroes of the quadratic polynomial h = 25t − 5t² are 0 and 5. Since 25t − 5t² = 5t(5 − t), h = 0 when t = 0 or t = 5. These are also the x-coordinates of the points where the parabola cuts the x-axis (the…

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7. Q7Case-based4 marksAsked: 2023

   Case Study - 3 38. In a pool at an aquarium, a dolphin jumps out of the water travelling at 20 cm per second. Its height above water level after t seconds is given by h = 20t - 16t^2. Based on the above, answer the following questions : (i) Find zeroes of polynomial p(t) = 20t - 16t^2. (ii) Which of the following types of graph represents p(t) ? (iii) (a) What would be the value of h at t = 3/2 ? Interpret the result. OR (iii) (b) How much distance has the dolphin covered before hitting the water level again ?

8. Q8Case-based4 marksAsked: 2026

   An arch of a railway bridge, built on Chenab riverbed, is shown in the above diagram. It is a parabolic arch connecting two hills at P and Q. If the parabolic curve is represented by the polynomial p(x) = -0.0025x^2 - 0.025x + 136. Observe the diagram and based on above information, answer the following questions : (i) Write the co-ordinates of point A. (1) (ii) Find the span of the arch. (1) (iii) (a) Write the zeroes of the polynomial using diagram and verify the relationship between sum of zeroes and polynomials. (2) OR (iii) (b) Find the values of p(x) at x = 100 and x = -100. Are they same ? (2)

9. Q9Case-based4 marksAsked: 2026

   An arch of a railway bridge, built on Chenab riverbed, is shown in the above diagram. It is a parabolic arch connecting two hills at P and Q. If the parabolic curve is represented by the polynomial p(x) = -0.0025x^2 - 0.025x + 136. Observe the diagram and based on above information, answer the following questions : (i) Write the co-ordinates of point A. 1 (ii) Find the span of the arch. 1 (iii) (a) Write the zeroes of the polynomial using diagram and verify the relationship between sum of zeroes and polynomials. 2 OR (iii) (b) Find the values of p(x) at x = 100 and x = -100. Are they same ? 2

10. Q10Case-based4 marksAsked: 2023

    In a pool at an aquarium, a dolphin jumps out of the water travelling at 20 cm per second. Its height above water level after t seconds is given by h = 20t - 16t^2. Based on the above, answer the following questions : (i) Find zeroes of polynomial p(t) = 20t - 16t^2. (ii) Which of the following types of graph represents p(t) ? (iii) (a) What will be the value of h at t = 3/2 ? Justify the answer. OR (iii) (b) What is the distance travelled by the dolphin before hitting the water level again ?

    How the model answer begins

    1. (i) p(t) = 20t − 16t² = 4t(5 − 4t) = 0.

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11. Q11Case-based4 marksAsked: 2023

    Rainbow is an arch of colours that is visible in the sky after rain or when water droplets are present in the atmosphere. The colours of the rainbow are generally, red, orange, yellow, green, blue, indigo and violet. Each colour of the rainbow makes a parabola. We know that any quadratic polynomial p(x) = ax^2 + bx + c (a != 0) represents a parabola on the graph paper. Based on the above, answer the following questions : (i) The graph of a rainbow y = f(x) is shown in the figure. Write the number of zeroes of the curve. (ii) If the graph of a rainbow does not intersect the x-axis but intersects y-axis at one point, then how many zeroes will it have ? (iii) (a) If a rainbow is represented by the quadratic polynomial p(x) = x^2 + (a + 1) x + b, whose zeroes are 2 and - 3, find the value of a and b. OR (iii) (b) The polynomial x^2 - 2x - (7p + 3) represents a rainbow. If - 4 is a zero of it, find the value of p.

    How the model answer begins

    1. (i) The curve has two zeroes. The graph of the rainbow y = f(x) is an arch, a parabola open downwards, and it cuts the x-axis at two distinct points, one on each side of its highest point. The zeroes of a polynomial…

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12. Q12Case-based4 marksAsked: 2023

    [passage] Based on the above, answer the following questions : (i) The graph of a rainbow y = f(x) is shown in the figure. Write the number of zeroes of the curve. 1 (ii) If the graph of a rainbow does not intersect the x-axis but intersects y-axis at one point, then how many zeroes will it have ? 1 (iii) (a) If a rainbow is represented by the quadratic polynomial p(x) = x^2 + (a + 1) x + b, whose zeroes are 2 and -3, find the value of a and b. 2 OR (iii) (b) The polynomial x^2 - 2x - (7p + 3) represents a rainbow. If -4 is a zero of it, find the value of p. 2

    How the model answer begins

    1. (i) The curve has two zeroes. The zeroes of a polynomial p(x) are precisely the x-coordinates of the points where the graph of y = p(x) intersects the x-axis. The rainbow curve intersects the x-axis at two points, so it…

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13. Q13Short answer3 marksAsked: 2025

    Find the zeroes of the polynomial p(x) = 3x^2 - 2x - 1 and verify the relationship between the zeroes of p(x) and the coefficients of p(x).

    How the model answer begins

    1. Factorising p(x) = 3x² – 2x – 1 by splitting the middle term: p(x) = 3x² – 3x + x – 1 = 3x(x – 1) + 1(x – 1) = (3x + 1)(x – 1)

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14. Q14Short answer3 marksAsked: 2025

    Find the zeroes of the polynomial p(x) = 2x^2 + 5x + 2 and verify the relationship between zeroes of p(x) and its coefficients.

15. Q15Short answer3 marksAsked: 2025

    Find the zeroes of the polynomial q(x) = 6x^2 - 5x - 1 and verify the relationship between the zeroes of q(x) and its coefficients.

    How the model answer begins

    1. Given: q(x) = 6x^2 - 5x - 1. Here the coefficient of x^2 is 6, the coefficient of x is -5 and the constant term is -1. By the method of splitting the middle term: 6x^2 - 5x - 1 = 6x^2 - 6x + x - 1 = 6x(x - 1) + 1(x - 1)…

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16. Q16Short answer3 marksAsked: 2025

    Find the zeroes of the polynomial 9s^2 - 6s + 1 and verify the relationship between the zeroes and the coefficients of the given polynomial.

    How the model answer begins

    1. We factorise the given polynomial 9s² – 6s + 1 by splitting the middle term. Here –6s is written as –3s – 3s, because (–3s) × (–3s) = 9s × 1 = 9s². 9s² – 6s + 1 = 9s² – 3s – 3s + 1 = 3s(3s – 1) – 1(3s – 1) = (3s – 1)(3s…

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17. Q17Short answer3 marksAsked: 2025

    Find the zeroes of the polynomial 25a^2 - 10a + 1 and verify the relationship between the zeroes and the coefficients of the given polynomial.

    How the model answer begins

    1. Given polynomial: 25a² – 10a + 1. By the method of splitting the middle term: 25a² – 10a + 1 = 25a² – 5a – 5a + 1 = 5a(5a – 1) – 1(5a – 1) = (5a – 1)(5a – 1)

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18. Q18Short answer3 marksAsked: 2025

    (a) If alpha, beta are zeroes of the polynomial 3x^2 - 8x + 4, then form a quadratic polynomial in x whose zeroes are 1/alpha and 1/beta. OR (b) Find zeroes of the polynomial 6x^2 - 7x - 3 and verify the relationship between zeroes and its coefficients.

    How the model answer begins

    1. (a) Given: p(x) = 3x^2 – 8x + 4 has zeroes α and β. Comparing it with ax^2 + bx + c, we get a = 3, b = –8, c = 4. Sum of zeroes: α + β = –b/a = –(–8)/3 = 8/3 Product of zeroes: αβ = c/a = 4/3

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19. Q19Short answer3 marksAsked: 2025

    (a) If alpha, beta are zeroes of the polynomial 8x^2 - 5x - 1, then form a quadratic polynomial in x whose zeroes are 2/alpha and 2/beta. OR (b) Find the zeroes of the polynomial p(x) = 3x^2 + x - 10 and verify the relationship between zeroes and its coefficients.

    How the model answer begins

    1. (a) Given: α and β are the zeroes of the quadratic polynomial p(x) = 8x² – 5x – 1. We compare it with ax² + bx + c: a = 8, b = –5, c = –1.

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20. Q20Short answer3 marksAsked: 2025

    (a) alpha, beta are zeroes of the polynomial 3x^2 - 8x + k. Find the value of k, if alpha^2 + beta^2 = 40/9. OR (b) Find the zeroes of the polynomial 2x^2 + 7x + 5 and verify the relationship between its zeroes and co-efficients.

    How the model answer begins

    1. Given: α and β are the zeroes of the quadratic polynomial p(x) = 3x² − 8x + k. Comparing it with ax² + bx + c gives a = 3, b = −8, c = k. Sum of zeroes: α + β = −b/a = −(−8)/3 = 8/3 Product of zeroes: αβ = c/a = k/3

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44 more questions of this chapter, each with its model answer, in the Ask Medha app.

Objective questions with the board’s key (10)

1. 1. CBSE 2026

   The graph of y = f(x) is given. The number of zeroes of f(x) is :

   • (A) 0 ✓
   • (B) 1
   • (C) 2
   • (D) 4

   Answer: (A)

2. 2. CBSE 2026

   Assertion (A) : The polynomial p(y) = y^2 + 4y + 3 has two zeroes. Reason (R) : A quadratic polynomial can have at most two zeroes.

   • (A) Both, Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of the Assertion (A).
   • (B) Both, Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A). ✓
   • (C) Assertion (A) is true, but Reason (R) is false.
   • (D) Assertion (A) is false, but Reason (R) is true.

   Answer: (B)

3. 3. CBSE 2026

   The graph of y = f(x) is given. The number of distinct zeroes of y = f(x) is :

   • (A) 0
   • (B) 1
   • (C) 2 ✓
   • (D) 3

   Answer: (C)

4. 4. CBSE 2026

   If alpha and beta are the zeroes of the polynomial f(x) = px^2 - 2x + 3p and alpha + beta = alpha beta, then the value of p is :

   • (A) - 2/3
   • (B) 2/3 ✓
   • (C) 1/3
   • (D) - 1/3

   Answer: (B)

5. 5. CBSE 2026

   If alpha and beta are two zeroes of a polynomial f(x) = px^2 - 2x + 3p and alpha + beta = alpha*beta, then value of p is :

   • (A) - 2/3
   • (B) 2/3 ✓
   • (C) 1/3
   • (D) - 1/3

   Answer: (B)

6. 6. CBSE 2026

   How many zeroes does p(x) = (x - 2) (x + 3) have ?

   • (A) Zero
   • (B) One
   • (C) Two ✓
   • (D) Three

   Answer: (C)

7. 7. CBSE 2026

   If alpha and beta are two zeroes of a polynomial f(x) = px^2 - 2x + 3p and alpha + beta = alpha beta, then value of p is :

   • (A) -2/3
   • (B) 2/3 ✓
   • (C) 1/3
   • (D) -1/3

   Answer: (B)

8. 8. CBSE 2026

   The sum and product of zeroes of a quadratic polynomial p(x) are -1/3 and 2 respectively. The polynomial p(x) is :

   • (A) 3x^2 - x + 6
   • (B) x^2 + 1/3x - 2
   • (C) 3x^2 - x + 2
   • (D) - 3x^2 - x - 6 ✓

   Answer: (D)

9. 9. CBSE 2026

   A polynomial p(x), which has sum of its zeroes equal to their product, is :

   • (A) 3x^2 + 2x + 2
   • (B) 3x^2 - 2x - 3
   • (C) 3x^2 - 2x + 2 ✓
   • (D) x^2 - 3x + 2

   Answer: (C)

10. 10. CBSE 2026

    The graph of polynomial p(x) = k is shown here. Number of zeroes of polynomial p(x) is :

    • (A) 0 ✓
    • (B) 1
    • (C) 2
    • (D) infinitely many

    Answer: (A)

Questions parents and students ask

How many marks does "Polynomials" carry in CBSE Class 10 Maths?

In the CBSE papers of 2023–2026, this chapter carried about 4 marks and about 3 questions a year; by marks it ranks 14 in Maths.

Which are the most important (VVI) questions of "Polynomials"?

The 20 questions above are from the board’s real papers, each with the years it was asked; the most-asked come first with their full model answers. All 64 written questions of this chapter, with answers, are in the Ask Medha app.

Do the answers follow the board’s marking scheme?

Yes. CBSE publishes its marking schemes; every model answer here is written point by point to the marks the board’s own scheme gives, and checked by a second, independent model.